Problem 40 - Calculus Early Transcendentals

Question

Show that the Second Derivative Test is inconclusive when applied to the following functions at \((0,0) .\) Describe the behavior of the function at the critical point. $$f(x, y)=x^{2} y-3$$

Step-by-Step Solution

Answer

Answer: The Second Derivative Test is inconclusive for the function \(f(x, y) = x^2y\) at the critical point \((0,0)\) because the determinant of the Hessian matrix at this point is zero. We cannot determine whether the function has a local maximum, minimum, or saddle point at \((0,0)\) using the Second Derivative Test.

Step 1: Calculate the first partial derivatives

Calculate the first partial derivatives of the function \(f(x, y)\) with respect to x and y: $$f_{x}=\frac{\partial f}{\partial x}=2xy$$ $$f_{y}=\frac{\partial f}{\partial y}=x^2$$

Step 2: Find the critical point

To find the critical points, set both first partial derivatives equal to zero and solve for x and y: $$2xy=0$$ $$x^2=0$$ From this, we can see that the only critical point is \((0,0)\).

Step 3: Calculate the second partial derivatives

Calculate the second partial derivatives of the function \(f(x, y)\) with respect to x and y: $$f_{xx}=\frac{\partial^2 f}{\partial x^2}=2y$$ $$f_{yy}=\frac{\partial^2 f}{\partial y^2}=0$$ $$f_{xy}=\frac{\partial^2 f}{\partial x \partial y}=2x$$ $$f_{yx}=\frac{\partial^2 f}{\partial y \partial x}=2x$$

Step 4: Evaluate the determinant of the Hessian matrix at the critical point

The Hessian matrix is given by: $$H=\begin{bmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{bmatrix}$$ Evaluate the determinant of the Hessian matrix at the critical point \((0,0)\): $$\text{det}(H(0,0))=\begin{vmatrix} 2(0) & 2(0) \\ 2(0) & 0 \end{vmatrix}=(0)(0)-(2(0))(2(0))=0$$

Step 5: Apply the Second Derivative Test and describe the behavior

Since the determinant of the Hessian matrix at the critical point \((0,0)\) is zero, the Second Derivative Test is inconclusive. It cannot determine whether the function has a local maximum, minimum, or saddle point at \((0,0)\). To describe the behavior at the critical point, we must consider other methods and analyze the partial derivatives.

Key Concepts

Partial Derivatives

Partial derivatives are a central concept in multivariable calculus. They represent the rate at which a function changes as one of the input variables changes, while other variables remain constant. For a function like \( f(x, y) = x^2 y - 3 \), partial derivatives help us understand how the function behaves when either \( x \) or \( y \) is varied individually.

• The partial derivative of \( f \) with respect to \( x \) is denoted by \( f_x \) and finds the slope of the tangent to the curve at a fixed value of \( y \), calculated as \( \frac{\partial f}{\partial x} = 2xy \).
• The partial derivative of \( f \) with respect to \( y \), \( f_y \), evaluates the change in \( f \) as \( y \) changes, given by \( \frac{\partial f}{\partial y} = x^2 \).

Understanding how these derivatives behave allows us to explore critical points where the gradient might suggest a maximum, minimum, or saddle point.

Hessian Matrix

The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function. It is used in multivariable calculus to assess the local curvature of a function. For our function, the Hessian matrix at any point \((x, y)\) is expressed by the matrix:\[H = \begin{bmatrix} f_{xx} & f_{xy} \ f_{yx} & f_{yy}\end{bmatrix}\]
At the critical point \( (0,0) \), where our partial derivatives become zero, the Hessian becomes:\[H(0,0) = \begin{bmatrix} 2y & 2x \ 2x & 0\end{bmatrix}\]

This specific arrangement of derivatives in the matrix plays a crucial role in using the second derivative test to classify critical points, by examining the determinant of the Hessian.

Critical Points

Critical points occur where the first partial derivatives of a function are zero or undefined. For the function \( f(x, y) = x^2 y - 3 \), the critical point is found by solving the equations:\[2xy = 0\]\[x^2 = 0\]
The solution \( (x, y) = (0, 0) \) is the only critical point. At this point, both partial derivatives are zero. However, identifying such points is just the beginning; determining their nature requires further analysis with techniques like the second derivative test.

Critical points can lead to many possibilities such as local maxima or minima, or a saddle point. The next step often involves using the Hessian matrix or alternative analysis methods to better understand the behavior at these critical loci.

Mathematical Analysis

Mathematical analysis provides the tools and methods by which we can better understand the behavior of functions, especially around critical points. In particular, the second derivative test examines the concavity of the function at a critical point using the Hessian determinant:

• If \( \text{det}(H) > 0 \) and \( f_{xx} > 0 \), the function has a local minimum.
• If \( \text{det}(H) > 0 \) and \( f_{xx} < 0 \), the function has a local maximum.
• If \( \text{det}(H) < 0 \), the critical point is a saddle point.
• If \( \text{det}(H) = 0 \), the test is inconclusive, and further analysis is needed.

In our case, with \( \text{det}(H(0,0)) = 0 \), the second derivative test does not provide sufficient information. Additional approaches, such as analyzing higher-order derivatives or alternative mathematical techniques, may help describe the function's behavior at the critical point \((0,0)\).